Abstract
We consider an undirected graph G=(VG,EG) with a set T⊆VG of terminals, and with nonnegative integer capacities c(v) and costs a(v) of nodes v∈VG. A path in G is a T-path if its ends are distinct terminals. By a multiflow we mean a function F assigning to each T-path P a nonnegative rational weight F(P), and a multiflow is called feasible if the sum of weights of T-paths through each node v does not exceed c(v). The value of F is the sum of weights F(P), and the cost of F is the sum of F(P) times the cost of P w.r.t. a, over all T-paths P.
Generalizing known results on edge-capacitated multiflows, we show that the problem of finding a minimum cost multiflow among the feasible multiflows of maximum possible value admits half-integer optimal primal and dual solutions. Moreover, we devise a strongly polynomial algorithm for finding such optimal solutions.
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Babenko, M.A., Karzanov, A.V. Min-cost multiflows in node-capacitated undirected networks. J Comb Optim 24, 202–228 (2012). https://doi.org/10.1007/s10878-011-9377-3
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DOI: https://doi.org/10.1007/s10878-011-9377-3